On the initial mass-radius relation of stellar clusters

TL;DR

This paper presents an analytic model explaining the mass-radius relation of stellar clusters, highlighting environmental effects and feedback processes that influence cluster formation and evolution across cosmic time.

Contribution

The model predicts a specific mass-radius relation influenced by gas surface density and feedback, aligning well with observations and explaining cluster compactness variations.

Findings

High-pressure environments lead to more compact clusters.

The model reproduces observed mass-radius relations across environments.

Globular clusters at high redshift were more compact, aiding their survival.

Abstract

Young stellar clusters across nearly five orders of magnitude in mass appear to follow a power-law mass-radius relationship (MRR), $R_{\star} \propto M_{\star}^{\alpha}$, with $\alpha \approx 0.2 - 0.33$. We develop a simple analytic model for the cluster mass-radius relation. We consider a galaxy disc in hydrostatic equilibrium, which hosts a population of molecular clouds that fragment into clumps undergoing cluster formation and feedback-driven expansion. The model predicts a mass-radius relation of $R_{\star} \propto M_{\star}^{1/2}$ and a dependence on the kpc-scale gas surface density $R_{\star} \propto \Sigma_{\rm g}^{-1/2}$, which results from the formation of more compact clouds (and cluster-forming clumps within) at higher gas surface densities. This environmental dependence implies that the high-pressure environments in which the most massive clusters can form also induce the formation of clusters with the smallest radii, thereby shallowing the observed MRR at high-masses towards the observed $R_{\star} \propto M_{\star}^{1/3}$. At low cluster masses, relaxation-driven expansion induces a similar shallowing of the MRR. We combine our predicted MRR with a simple population synthesis model and apply it to a variety of star-forming environments, finding good agreement. Our model predicts that the high-pressure formation environments of globular clusters at high redshift naturally led to the formation of clusters that are considerably more compact than those in the local Universe, thereby increasing their resilience to tidal shock-driven disruption and contributing to their survival until the present day.